In physics and particularly in particle physics, a multiplet is the state space for 'internal' degrees of freedom of a particle; that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as the particle's position in space. Examples of such degrees of freedom are the spin state of a particle in quantum mechanics, or the color, isospin and hypercharge state of particles in the Standard Model of particle physics. Formally, we describe this state space by a vector space which carries the action of a group of continuous symmetries.

Mathematically, multiplets are described via representations of a Lie group or its corresponding Lie algebra, and is usually used to refer to irreducible representations (irreps, for short).

At the group level, this is a triplet ${\displaystyle (V,G,\rho )}$ where

At the algebra level, this is a triplet ${\displaystyle (V,{\mathfrak {g}},\rho )}$, where

The symbol ${\displaystyle \rho }$ is used for both Lie algebras and Lie groups as, at least in finite dimension, there is a well understood correspondence between Lie groups and Lie algebras.

In mathematics, it is common to refer to the homomorphism ${\displaystyle \rho }$ as the representation, for example in the sentence 'consider a representation ${\displaystyle \rho }$', and the vector space ${\displaystyle V}$ is referred to as the 'representation space'. In physics sometimes the vector space is referred to as the representation, for example in the sentence 'we model the particle as transforming in the singlet representation', or even to refer to a quantum field which takes values in such a representation, and the physical particles which are modelled by such a quantum field.

For an irreducible representation, an ${\displaystyle n}$-plet refers to an ${\displaystyle n}$ dimensional irreducible representation. Generally, a group may have multiple non-isomorphic representations of the same dimension, so this does not fully characterize the representation. An exception is ${\displaystyle {\text{SU}}(2)}$ which has exactly one irreducible representation of dimension ${\displaystyle n}$ for each non-negative integer ${\displaystyle n}$.

For example, consider real three-dimensional space, ${\displaystyle \mathbb {R} ^{3}}$. The group of 3D rotations SO(3) acts naturally on this space as a group of ${\displaystyle 3\times 3}$ matrices. This explicit realisation of the rotation group is known as the fundamental representation ${\displaystyle \rho _{\text{fund}}}$, so ${\displaystyle \mathbb {R} ^{3}}$ is a representation space. The full data of the representation is ${\displaystyle (\mathbb {R} ^{3},{\text{SO(3)}},\rho _{\text{fund}})}$. Since the dimension of this representation space is 3, this is known as the triplet representation for ${\displaystyle {\text{SO}}(3)}$, and it is common to denote this as ${\displaystyle \mathbf {3} }$.